a) [latex](a^{3})^{-2} * (a^{-2})^{-4} = a^{-6} * a^{8} = a^{2} [/latex] b)[latex](a^{3})^{-5} * (a^{4})^{-4} = a^{-15} * a^{-16} = a^{-31} [/latex] c)[latex](a^{6})^{2} : (a^{-4})^{-5} = a^{12} : a^{20} = a^{-8} [/latex] d)[latex](a^{-5})^{3} : (a^{-6})^{-2} = a^{-15} : a^{12} = a^{-27} [/latex] e)[latex] frac{(a^{3} * a^{-5}) : a^{-4}}{a^{-3} * a^{6}} = frac{a^{-2}: a^{-4}}{a^{3}} = frac{a^{2}}{a^{3}} = a^{-1}[/latex] f)[latex] frac{a^{-2} * a^{-6} * a^{5}}{(a^{4})^{-2} * (a^{-3})^{-5}} = frac{a^{-3}}{a^{-8} * a^{15}} = frac{a^{-3}}{a^{7}} = a^{-10}[/latex] g)[latex] frac{(a^{6})^{-3} : (a^{4})^{-5}}{(a^{-3})^{-3} * (a^{2})^{-6}} = frac{a^{-18} : a^{-20}}{a^{9} * a^{-12}} = frac{a^{2}}{a^{-3}} = a^{5} [/latex] h)[latex] frac{[(a^{3})^{-5} * a^{-2}] : a^{-4}}{[(a^{-2})^{-5} : a^{-6}] * a^{2}} = frac{[a^{-15}* a^{-2}] : a^{-4}}{[a^{10} : a^{-6}] * a^{2}} = frac{a^{-17} : a^{-4}}{a^{16} * a^{2}} = frac{a^{-13}}{a^{18}} = a^{-31} [/latex] i)[latex][ frac{a^{4} * a^{-6} }{a^{-5} : a^{-3}} ]^{-2} = [ frac{a^{-2}}{a^{-2}} ]^{-2} = [ a^{0}]^{-2} = a^{0} [/latex] j)[latex][a^{-6} * frac{(a^{-3})^{-5}}{a^{2}}]^{-3} = [a^{-6} * a^{13}]^{-3} = [a^{7}]^{-3} = a^{-21}[/latex]
